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Non-Degenerate Perturbation Theory
Let us now generalize our perturbation analysis to deal with systems
possessing more than two energy eigenstates. The energy eigenstates of the
unperturbed Hamiltonian,
, are denoted
![$\displaystyle H_0\, \vert n\rangle = E_n\, \vert n\rangle,$](img1775.png) |
(7.19) |
where
runs from 1 to
. The eigenkets
are orthonormal, and
form a complete set.
Let us now try to solve the energy eigenvalue
problem for the perturbed Hamiltonian:
![$\displaystyle (H_0 + H_1) \,\vert E\rangle = E\,\vert E\rangle.$](img1750.png) |
(7.20) |
We can express
as a linear superposition of the unperturbed energy
eigenkets,
![$\displaystyle \vert E\rangle = \sum_k \langle k \vert E\rangle \vert k\rangle,$](img1776.png) |
(7.21) |
where the summation is from
to
. Substituting the previous
equation into Equation (7.20), and right-multiplying by
, we obtain
![$\displaystyle (E_m + e_{mm} - E)\, \langle m\vert E\rangle + \sum_{k\neq m} e_{mk}\, \langle k\vert E\rangle = 0,$](img1779.png) |
(7.22) |
where
![$\displaystyle e_{mk} = \langle m \vert\,H_1\,\vert k\rangle.$](img1780.png) |
(7.23) |
Let us now develop our perturbation expansion. We assume that
![$\displaystyle \frac{\vert e_{mk}\vert}{E_m - E_k} \sim {\cal O}(\epsilon),$](img1781.png) |
(7.24) |
for all
, where
is our expansion parameter. We also
assume that
![$\displaystyle \frac{\vert e_{mm}\vert}{E_m} \sim {\cal O}(\epsilon),$](img1784.png) |
(7.25) |
for all
. Let us search for a modified version of the
th unperturbed energy
eigenstate for which
![$\displaystyle E= E_n + {\cal O}(\epsilon),$](img1785.png) |
(7.26) |
and
for
. Suppose that we
write out Equation (7.22) for
, neglecting terms that
are
according to our expansion scheme. We find that
![$\displaystyle (E_m - E_n) \,\langle m \vert E \rangle + e_{mn} \simeq 0,$](img1791.png) |
(7.29) |
giving
![$\displaystyle \langle m\vert E\rangle \simeq -\frac{e_{mn}}{E_m - E_n}.$](img1792.png) |
(7.30) |
Substituting the previous expression into Equation (7.22),
evaluated for
, and neglecting
terms, we obtain
![$\displaystyle (E_n + e_{nn} - E) - \sum_{k\neq n} \frac{\vert e_{nk}\vert^{\,2}} {E_k-E_n} \simeq 0.$](img1795.png) |
(7.31) |
Thus, the modified
th energy eigenstate possesses the eigenvalue
![$\displaystyle E_n' = E_n + e_{nn} + \sum_{k\neq n} \frac{\vert e_{nk}\vert^{\,2}} {E_n-E_k} + {\cal O}(\epsilon^{\,3}),$](img1796.png) |
(7.32) |
and the eigenket
![$\displaystyle \vert n\rangle' = \vert n\rangle +\sum_{k\neq n}\frac{e_{kn}}{E_n - E_k}\,\vert k\rangle + {\cal O}(\epsilon^{\,2}).$](img1797.png) |
(7.33) |
Note that
![$\displaystyle \langle m\vert n\rangle' = \delta_{mn} + \frac{e_{nm}^{\,\ast}}{E...
... {E_n-E_m} + {\cal O}(\epsilon^{\,2}) = \delta_{mn} + {\cal O}(\epsilon^{\,2}).$](img1798.png) |
(7.34) |
Thus, the modified eigenkets remain orthonormal
to
.
Note, finally, that if the perturbing Hamiltonian,
, commutes with the unperturbed Hamiltonian,
, then
![$\displaystyle e_{ml}= e_{mm}\,\delta_{ml},$](img1799.png) |
(7.35) |
and
The previous two equations are exact (i.e., they hold to all orders in
).
Next: Quadratic Stark Effect
Up: Time-Independent Perturbation Theory
Previous: Two-State System
Richard Fitzpatrick
2016-01-22